Physics | Forces and momentum

Welcome to A.2: Forces and Momentum!

Hello future physicists! This chapter, Forces and Momentum, is the heart of classical mechanics. If A.1 (Kinematics) told us *how* things move (velocity, acceleration), A.2 tells us *why* they move—it introduces the concept of Force and its effect on motion, which we quantify using Momentum.

Understanding this section is vital because forces are responsible for everything, from the pull of gravity keeping you on Earth to the tiny pushes that make a tennis ball fly. Don'b t worry if some concepts, like impulse, seem abstract; we'll break them down using plenty of real-world examples!

1. Newton's Laws: The Foundation of Dynamics

Sir Isaac Newton laid down three fundamental laws that govern how objects interact with forces. Mastering these is non-negotiable for success in Physics.

1.1. Newton's First Law (N1L): The Law of Inertia

An object continues in a state of rest or uniform velocity unless acted upon by a net (unbalanced) external force.

• Key Term: Inertia. This is the property of an object that resists changes in its state of motion. The more mass an object has, the greater its inertia.
• If the Net Force (\(F_{net}\)) acting on an object is zero, the object is in translational equilibrium. This means:
  • If it was resting, it stays resting.
  • If it was moving, it continues moving at a constant velocity (constant speed, straight line).

Real-World Example: When your car brakes suddenly, your body keeps moving forward due to inertia (until the seatbelt or dashboard applies a net force to stop you).

1.2. Newton's Second Law (N2L): Force, Mass, and Acceleration

This law provides the mathematical link between force and motion. It states that the net force acting on an object is equal to the rate of change of its momentum, and is proportional to its mass and acceleration.

The Definition Formula:
$$F_{net} = \frac{\Delta p}{\Delta t}$$

The Calculation Formula (for constant mass):
$$F_{net} = ma$$

• \(F_{net}\): The vector sum of all external forces acting on the object (measured in Newtons, N).
• \(m\): The mass of the object (kg).
• \(a\): The resulting acceleration (m s\(^{-2}\)).

Important Takeaway: Force and acceleration are always in the same direction! If you push North, the object accelerates North.

Did you know? N2L is arguably the most used equation in classical mechanics. It tells us that a small force can accelerate a large mass slowly, or a large force can accelerate a small mass quickly.

1.3. Newton's Third Law (N3L): Action-Reaction

Whenever one object exerts a force on a second object (the action force), the second object simultaneously exerts an equal and opposite force on the first object (the reaction force).

• Forces always occur in pairs.
• The forces are equal in magnitude and opposite in direction.
• Crucially, the forces in an N3L pair act on different bodies. They can never cancel each other out because they don't act on the same system.

Analogy: When you push against a wall, the wall pushes back on you with exactly the same force. If you push harder, the wall pushes back harder.

2. Free-Body Diagrams (FBDs) and Force Components

2.1. The Importance of FBDs

A Free-Body Diagram (FBD) is a schematic tool essential for solving force problems. It isolates the object of interest and shows all the external forces acting *on* it as vectors originating from the center of mass.

Step-by-Step Guide to Drawing an FBD:

1. Isolate the body (represent it as a dot or simple block).
2. Identify all forces acting *on* the body (e.g., weight, normal, tension, friction, applied force).
3. Draw vector arrows for each force, clearly indicating their direction.
4. Resolve forces into components if they are not parallel or perpendicular to the direction of motion (usually x and y axes, or parallel/perpendicular to an incline).

2.2. Standard Forces

• Weight (\(W\) or \(F_g\)): The force of gravity acting on a mass. Always vertically downwards. $$W = mg$$
• Normal Force (\(N\) or \(F_N\)): The force exerted by a surface perpendicular to that surface to prevent objects from passing through it.
• Tension (\(T\)): Force exerted by a string, rope, or cable. Always pulls away from the object, parallel to the rope.

2.3. Force of Friction

Friction (\(f\)) is a resistive force that opposes motion. It acts parallel to the surface.

Static Friction (\(f_s\))

This is the friction that prevents an object from starting to move. It adjusts its magnitude up to a maximum value (\(f_{s, max}\)).

$$f_{s, max} = \mu_s N$$

• \(\mu_s\) is the coefficient of static friction.
• Motion *begins* when the applied force exceeds \(f_{s, max}\).

Dynamic (Kinetic) Friction (\(f_d\) or \(f_k\))

This is the friction that acts on an object while it is already moving. It is usually constant.

$$f_d = \mu_d N$$

• \(\mu_d\) is the coefficient of dynamic friction.
• For almost all surfaces, \(\mu_s > \mu_d\). This is why it is harder to get a heavy box moving than it is to keep it moving!

Key Takeaway: Static friction has a variable magnitude up to its maximum; dynamic friction usually has a constant magnitude once motion starts.

3. Linear Momentum (\(p\))

Momentum is often described as "mass in motion." It is a fundamental property in physics that describes the difficulty of stopping a moving object.

3.1. Definition of Momentum

Linear Momentum (\(p\)) is defined as the product of an object's mass and its velocity.

$$p = mv$$

• Momentum is a vector quantity (it has magnitude and direction). The direction of \(p\) is the same as the direction of the velocity, \(v\).
• The SI unit for momentum is kg m s\(^{-1}\) (kilogram meters per second).

Analogy: Imagine trying to stop a bowling ball (large mass, medium velocity) versus a bullet (small mass, huge velocity). Both have significant momentum, making them hard to stop!

3.2. Impulse (\(I\))

Impulse quantifies the overall effect of a force applied over a period of time. It is the measure of the "oomph" delivered by a force.

$$I = F \Delta t$$

• \(F\) is the net force applied (N).
• \(\Delta t\) is the duration over which the force acts (s).
• The SI unit for Impulse is N s (Newton-seconds).

3.3. The Impulse-Momentum Theorem

This theorem directly links the concepts of impulse and momentum, showing that the impulse applied to an object is equal to the change in its momentum.

$$I = \Delta p$$ $$F \Delta t = \Delta p = mv_{final} - mv_{initial}$$

This theorem is incredibly powerful because it relates microscopic events (a collision) to macroscopic variables (force and time).

Real-World Safety Connection: Why do airbags and crumple zones save lives?
They work by increasing the time of impact (\(\Delta t\)) during a collision. Since the required change in momentum (\(\Delta p\)) is fixed (you must stop!), increasing \(\Delta t\) means the average force (\(F\)) exerted on the passenger is greatly reduced. $$F = \frac{\Delta p}{\Delta t}$$

4. Conservation of Linear Momentum

4.1. The Principle

In an isolated system (a system where the net external force is zero, meaning only internal forces like collisions are acting), the total linear momentum remains constant.

Law of Conservation of Linear Momentum:
$$p_{total, initial} = p_{total, final}$$
For two objects (A and B): $$m_A u_A + m_B u_B = m_A v_A + m_B v_B$$

• Crucial Requirement: This law only applies if the system is closed (no mass added or removed) and isolated (\(F_{net, external} = 0\)).
• Because momentum is a vector, this conservation must hold true independently for the x-direction and the y-direction.

4.2. Collisions and Explosions

The principle of conservation of momentum applies to all interactions, including collisions (objects joining or bouncing apart) and explosions (objects starting together and moving apart).

Types of Collisions

Collisions are typically classified based on what happens to Kinetic Energy (KE) during the event. Momentum is always conserved in all collisions, provided the system is isolated.

1. Elastic Collision:
   • Both Momentum AND Kinetic Energy are conserved.
   • Objects bounce off each other without permanent deformation or energy loss to heat/sound. (Example: idealized billiard balls).
2. Inelastic Collision:
   • Momentum is conserved, but Kinetic Energy is NOT conserved (it is converted into heat, sound, or internal deformation).
   • Objects usually separate after impact. (Example: a car crash where KE is lost to crumpled metal and noise).
3. Perfectly Inelastic Collision:
   • The maximum possible KE is lost consistent with momentum conservation.
   • The objects stick together after the collision and move with a common final velocity. (Example: A dart sticking to a block of wood).

Memory Trick: To remember the difference: "E"lastic means "E"nergy is conserved. If it's Inelastic, energy is lost!